Nodal solutions of 2-D criticalnonlinear Schrödinger equationswith potentials vanishing at infinity

Mingwen Fei; Huicheng Yin

doi:10.3934/dcds.2015.35.2921

Article Contents

2015, Volume 35, Issue 7: 2921-2948. Doi: 10.3934/dcds.2015.35.2921

This issue Previous Article Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures Next Article Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions

Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity

Mingwen Fei^1, and
Huicheng Yin^1,

1.
Department of Mathematics & IMS, Nanjing University, Nanjing, 210093

Received: April 30, 2013

Revised: September 30, 2014

Published: May 2017

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Abstract

We will focus on the existence and concentration of nodal solutions to the following critical nonlinear Schrödinger equations in $\Bbb R^2$ $$ -\epsilon^2\triangle u_{\epsilon}+V(x)u_{\epsilon}=K(x) |u_{\epsilon}|^{p-2}u_{\epsilon}e^{\alpha_{0}|u_{\epsilon}| ^{2}},\quad u_{\epsilon}\in H^1(\Bbb R^2), $$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, and $\epsilon>0$ is a small constant. For the positive potential $V(x)$ which decays at infinity like $(1+|x|)^{-\alpha}$ with $0 < \alpha \le 2$, we will show that a nodal solution with one positive and one negative peaks exists, and concentrates around local minimum points of the related ground energy function $G(\xi)$ of the Schrödinger equation $ -\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{2}}$.

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Mathematics Subject Classification: Primary: 35B33, 35J60; Secondary: 35Q55.

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